Regular trends оf acousto-optic interaction in terahertz region оf
electromagnetic radiation
Pavel Nikitin1, Vitaly Voloshinov1, Vasiliy Gerasimov2, Boris Knyazev3
1 – Department of Physics, Lomonosov Moscow State University, 119991
Moscow, Russia
2 – Budker Institute of Nuclear Physics, Siberian Branch, Russian Academy of
Sciences, 630090 Novosibirsk, Russia
3 – Novosibirsk National Research State University, 630090 Novosibirsk,
Russia
1 – e-mail: nikitin.pavel.a@gmail.com
Basic problems of acousto-optic interaction in terahertz region of
electromagnetic spectrum are considered. We obtained experimental results that are
fitted by generalized theoretical model developed with the goals to describe the
interaction in details. Our analysis showed that crystalline germanium is one of the
best materials to observe the acousto-optic interaction in the terahertz range. The
carried out study proved that a germanium based acousto-optic device could bе used
for fast and reliable deflection of monochromatic radiation in the terahertz spectral
region.
PACS numbers: 78.20.Hp; 42.25.Fx
1. Introduction
Acousto-optic (AO) interaction is widely used to control such parameters of
electromagnetic radiation as direction of propagation, phase, frequency, intensity and
state of polarization [1-3]. Since nature of the phenomenon does not depend on
1
frequency of the electromagnetic radiation, AO devices have already been used in the
ultraviolet, visible, infrared and even in the terahertz (THz) spectral ranges. Usually,
these devices are AO modulators, deflectors and tunable filters. However, so far there
are no publications describing these devices and their operation in the THz range.
Unfortunately, a number of papers related to the AO interaction in the THz frequency
band оf the electromagnetic radiation is limited to a few contributions [1-2]. The main
reason for this drawback is a number of disappointing features of the AO interaction
in the THz region. These features namely are the low diffraction efficiency and the
substantial divergence of light, i.e. the THz radiation, due to its relatively long
wavelengths 𝜆 [3-5]. The second characteristic feature is related to the strong
absorption of light at the values of the absorption coefficient 𝛼 = 4𝑛′′/𝜆 of the order
from 1.0 to 10 cm-1. Here 𝑛′′ is the imaginary part of the refractive index 𝑛 = 𝑛′ −
𝑖𝑛′′, which is usually much smaller than the real part of it 𝑛′′ ≪ 𝑛′. The purpose of
our study was generalization of the well-known theory of the AO interaction over
media in part transparent in the THz region. Experimental examination of the AO
interaction in solid and liquid media promising for the AO applications in the THz
region was also one of the goals of the research.
2. Theory
We follow general approach of the plane-wave theory of the AO interaction [3]
to formulate the total electric field 𝑬 in terms of complex amplitudes of the diffracted
𝐶1 and transmitted 𝐶0 light waves:
𝑬(𝑥, 𝑦) = ∑𝑝=0,1 𝐶𝑝 (𝑥 ) exp[𝑖(𝒌𝒑 𝒓 − 𝜔𝑝 𝑡)],
(1)
where 𝒌𝒑 = 𝑘{cos 𝜃𝑝 , sin 𝜃𝑝 , 0} is the wave vector and 𝜔 is the angular frequency
of light and 𝑥 – the coordinate across the sound beam. In our analysis, changes in the
wave number 𝑘 are neglected. Note that Eq.(1) has the essential limitation in the
2
direction of propagation 𝜃𝑝 ≪ 1, which is satisfied only at the quasi-orthogonal and
the quasi-collinear geometry of AO interaction.
It is known that a plane acoustic wave changes permittivity 𝜀 of a medium:
⃗⃗ 𝒓
⃗ − Ω𝑡)
𝜀 = 𝑛′2 − 2𝑖𝑛′ 𝑛′′ + 2𝑛′∆𝑛′cos(𝑲
(2)
where 𝑲 is the acoustic wave vector which is equal to 𝐾 {1,0,0}, at the quasi-collinear
geometry, and 𝐾{0,1,0}, at the quasi-orthogonal geometry.
In order to derive a system of coupled wave equations, one has to substitute
Eq.(2) in the Maxwell equations and collect terms with equal time dependent
exponential factors. This procedure is possible only if the angular frequencies satisfy
the condition 𝜔𝑛 = 𝜔0 + 𝑛Ω. Bragg matching condition is evaluated by means of the
mismatch parameter ∆𝒌 = 𝒌𝟏 − 𝒌𝟎 − 𝑲, dependent only on 𝑥-component [3].
Therefore, we obtain the following equations describing the AO interaction in Bragg
regime:
𝜕𝐶 ⁄𝜕𝑥 = 0.5[−𝛼0 𝐶0 + 𝑖𝑞0 𝐶1 exp(𝑖∆𝑘𝑥)]
{ 0
,
𝜕𝐶1 ⁄𝜕𝑥 = 0.5[−𝛼1 𝐶0 + 𝑖𝑞1 𝐶1 exp(−𝑖∆𝑘𝑥)]
(3)
where 𝛼𝑝 = 2𝑘𝑝 𝑛′′/𝑛′ cos 𝜃𝑝 is the absorption coefficient and 𝑞𝑝 = 𝑘𝑝 ∆𝑛′ /𝑛′ cos 𝜃𝑝
is the well-known coupling coefficient.
Analytical solution of the system Eq.(3) with the boundary conditions 𝐶0 (0) =
1 and 𝐶1 (0) = 0 is as follows:
𝑠𝐿
𝐶0 (𝐿) = [cos ( 2 ) −
𝐶1 (𝐿) = 𝑖
𝑞1
𝑠
𝑖∆𝑘+𝛼0 −𝛼1
𝑠
𝑠𝐿
(2𝑖∆𝑘−𝛼0 −𝛼1 )𝐿
𝑠𝐿
sin ( 2 )] exp [
sin ( 2 ) exp [−
(2𝑖∆𝑘+𝛼0 +𝛼1 )𝐿
4
4
].
]) ,
(4)
(5)
where 𝑠 = √𝑞1 𝑞1 +∆𝑘 2 + (𝛼1 − 𝛼0 )(4𝑖∆𝑘 + 𝛼0 − 𝛼1 )/2 and 𝐿 is a width of the
sound beam usually equal to a length of a piezoelectric transducer.
Evaluation of the obtained expressions Eq.(4) and (5) is rather complicated.
However, if the angles 𝜃0 and 𝜃1 are narrow and the acoustic angular frequency is
3
much smaller than that of light then 𝛼𝑝 = 𝛼 = 2𝑘0 𝑛′′/𝑛′ and 𝑞𝑝 = 𝑞 =
𝑘0 ∆𝑛 ′
𝑛′
. Consequently, the derived expressions Eq. (4) and (5) become simpler:
𝑠𝐿
𝐶0 (𝐿) = [cos ( 2 ) − 𝑖
𝑞
∆𝑘
𝑠
𝑠𝐿
sin ( 2 )] exp [
(𝑖∆𝑘−𝛼)𝐿
2
(−𝑖∆𝑘−𝛼)𝐿
𝑠𝐿
𝐶1 (𝐿) = 𝑖 𝑠 sin ( 2 ) exp [
2
] ,
],
(6)
(7)
where 𝑠 is defined as 𝑠 = √𝑞 2 +∆𝑘 2 .
It is obvious that the absorption of light results in a decrease of the optimal length
of the AO interaction 𝐿opt required reaching maximum of the diffraction efficiency.
Here we consider the AO interaction between a plane electromagnetic wave incident
on ultrasound at Bragg incidence (∆𝑘 = 0) and a sound beam having constant acoustic
power. In case of the orthogonal AO interaction geometry, we have 𝑞 = 2𝐴/√𝐿,
where 𝐴 = 𝑐𝑜𝑛𝑠𝑡. In case of the collinear geometry, 𝑞 remains constant 𝑞 = 2𝐵,
where 𝐵 = 𝐴/√𝑙 and 𝑙 is a size of the acoustic beam in the direction orthogonal to the
plane of the AO interaction. We found that the intensity of the diffracted light 𝐼1 =
|𝐶1 |2 in the quasi-orthogonal and the quasi-collinear interaction is given by the
following equation in terms of the dimensionless parameters 𝑋1 = 𝛼/𝐴2, 𝑌1 = 𝐴2 𝐿1 ,
𝑋2 = 𝛼/𝐵 and 𝑌2 = 𝐵𝐿2 :
𝐼1 ort = 𝑒 −𝑋1 𝑌1 sin2 √𝑌1 , 𝐼1 col = 𝑒 −𝑋2 𝑌2 sin2 𝑌2 .
(8)
Now we fix the dimensionless absorption coefficient 𝑋 to find an optimal value
𝑌 𝑜𝑝𝑡 of the dimensionless interaction length 𝑌. We proved that an analytical solution
𝑌 𝑜𝑝𝑡 (𝑋) can be found only in the case of low diffraction efficiency, when 𝑌 ≪ 1,
opt
namely 𝑌1
opt
= 1/𝑋1 and 𝑌2
= 2/𝑋2. It is clear that numerical approach is
applicable for arbitrary values of the parameter 𝑋. The results of numerical
calculations are plotted in Fig. 1 in log-log scale.
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We managed to reveal correlation between the optimal length 𝐿𝑜𝑝𝑡 and the
𝑜𝑝𝑡
absorption coefficient 𝛼 in the case of a low diffraction efficiency 𝐿1
= 1/𝛼 and
opt
𝐿2 = 2/𝛼. The relation between 𝐿𝑜𝑝𝑡 and 𝛼 provides estimation of the optimal size
of the crystal even if the coupling coefficient is unknown. That is why, the intensities
of diffracted light are given by 𝐼1 ort = 𝐴2 /𝛼𝑒 and 𝐼1 col = 4(𝐵/𝛼𝑒)2 = 𝐼1 ort 4/𝛼𝑙𝑒.
We used these expressions to evaluate the efficiency of the AO interaction in the
terahertz region of spectrum.
3. Experimental results
We measured transparency of а few АО materials in the THz region of spectrum
in а preliminary experiment carried out by means of the Novosibirsk free-electron
laser (FEL) [4]. It was found that the majority of known AO-materials were opaque.
Therefore, basic experiments were carried out in AO cell fabricated of а high-resistive
single crystal germanium [5].
The scheme of the experimental setup is shown in the Fig. 2. Polarization and
aperture of the THz FEL radiation (1) was set by the polarizer (2) and the diaphragm
(3). The radiation passed through the AO cell (4) and it was registered by the Golay
detector GC-1T (5). Electrical signal at the RF-input of the AO cell was formed by the
pulse generator (6) and the high-frequency generator (7). The lock-in amplifier (8) was
applied to extract and amplify a diffracted signal coming from the Golay detector. The
experimental data was collected and processed using the personal computer (9).
The germanium crystal was chosen for the analysis due to its high refractive
index 𝑛 = 4 and relatively low absorption coefficient 𝛼 = 0.75 1/cm. Unfortunately,
in our experiment, the length of the piezoelectric transducer 𝐿 = 2.0 cm was longer
than the optimal length 𝐿opt = 1.3 cm. We measured two different dependences in the
carried out experiments [5]. The goal of the first experiment was related to
5
measurement of the frequency dependence of Bragg angle 𝜃(𝐹) in the cell. We
registered the dependence in the acoustic frequency band 25-38 MHz. The second
experiment gave evaluation of the acoustic frequency bandwidth of diffraction ∆𝐹
registered at а fixed angle 𝜃 of the radiation incidence. It was proved that the examined
device could bе used as а reliable deflector providing scanning of the monochromatic
THz radiation at the wavelength 𝜆 = 140 microns. The diffraction efficiency 𝜉 =
𝐼1 (𝐿)/𝐼0 (𝐿) was equal (5.0 ± 0.3) × 10−4 at 1.0 W of the driving electrical power.
Spatial resolution of the deflection was evaluated as 𝑁 = ∆𝐹𝜏 = 8 pixels, where the
quick-action of the device was about 𝜏 = 2 µs, and 3 dB acoustic frequency
bandwidth was ∆𝐹 = 4 MHz at the central frequency of about 30 MHz. It should be
noted that the registered value of the diffraction efficiency correlated with the general
theoretical estimation.
As for a more detailed information on the experimental
invesigation as well as on a graphical representation of the obtained results, they may
be found in the reference [5].
In the most resent experiments, we examined the АО interaction in a number of
non-polаr liquids such as cyclo-hexane 𝐶6 𝐻12 , white spirit 𝑊𝑆 and tetra-chloromethane 𝐶𝐶𝑙4 . The study proved the general trend that the efficiency of the AO
diffraction in germanium was one order higher than that in the liquids: 𝜉 = (2.7 ±
0.2) × 10−5 for 𝐶6 𝐻12 and 𝜉 = (5.0 ± 1.2) × 10−5 for 𝑊𝑆. Moreover, the acoustooptic deflection of light in 𝐶𝐶𝑙4 was not detected at all. We note that the large values
of experimental errors originated from intensity fluctuations of the THz radiation at
the operating frequencies of about 10-15 Hz. We had to use these extremely low
operating frequencies because they corresponded to maximal sensitivity of the used
Golay detector GC-1T [6].
4. Conclusion
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To summarize the foregoing, we state that the AO interaction in the THz range
of electromagnetic spectrum is rather weak and difficult to observe due to the strong
absorption and divergence of radiation. More sophisticated theoretical and
experimental investigations should be performed to understand principal aspects of
the diffraction phenomenon in the THz region. It is expected that further investigations
will result in development of new and efficient methods to control parameters of the
THz radiation. Another direction of future investigations is related to search of new
acousto-optic materials having high diffraction efficiency in the THz range. As for the
crystalline germanium, for today it seems that it is the best available material to realize
the acousto-optic interaction in the terahertz spectral range.
The research was supported by the grant N 14-12-00380 of the Russian
Scientific Fund.
7
[1] T. Vogel, G. Dodel, Infrared Phys., 25, 315 (1985). DOI: 10.1016/00200891(85)90097-1
[2] W. Durr, W. Schmidt, Int. J. of Infrared and Millimeter Waves, 6, 1043 (1985).
DOI: 10.1007/BF01010680
[3] V.I. Balakshy, V.N. Parygin and L.E. Chirkov, Physical Principles of AcoustoOptics, Radio i Svyaz, Moscow, 1982.
[4] B.A. Knyazev, G.N. Kulipanov and N.A. Vinokurov, Meas. Sci. Technol., 21,
054017 (2010). DOI: 10.1088/0957-0233/21/5/054017
[5] V.B. Voloshinov, P.A. Nikitin, V.V. Gerasimov, B.A. Knyazev and Yu.Yu.
Choporova,
Quantum
Electronics,
43,
10.1070/QE2013v043n12ABEH015195
[6] www.tydexoptics.com
8
1139
(2013).
DOI:
Figure 1. Dependence of the optimal dimensionless length of AO interaction on
the dimensionless absorption coefficient. The solid line corresponds to the
orthogonal geometry, while the dash line describes the collinear geometry.
Figure 2. Scheme of the experimental setup: (1) THz free-electron laser; (2)
polarizer; (3) diaphragm; (4) AO deflector; (5) Golay detector; (6) pulse
generator; (7) high-frequency generator; (8) lock-in amplifier and (9) personal
computer.
9
Figure 1
Figure 2
10